Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems

Transcription

1 Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems Michael J. eely arxiv: v3 math.oc 3 Dec 008 Abstract We consider the delay properties of max-weight opportunistic scheduling in a multi-user O/OFF wireless system, such as a multi-user downlin or uplin. It is well nown that max-weight scheduling stabilizes the networ (and hence yields maximum throughput) whenever input rates are inside the networ capacity region. We show that when arrival and channel processes are independent, average delay of the maxweight policy is order-optimal, in the sense that it does not grow with the number of networ lins. While recent queuegrouping algorithms are nown to also yield order-optimal delay, this is the first such result for the simpler class of max-weight policies. We then consider multi-rate transmission models and show that average delay in this case typically does increase with the networ size due to queues containing a small number of residual pacets. I. ITRODUCTIO We consider the delay properties of max-weight opportunistic scheduling in a multi-user wireless system. Specifically, we consider a system with transmission lins. Each lin receives independent data that arrives randomly and must be queued for eventual transmission. Separate queues are maintained by each lin i {,...,}, so that data arriving to queue i must be transmitted over lin i. The system wors in slotted time with normalized slots t {0,,,...}. The channel states of each lin vary randomly from slot to slot, and every slot t the networ controller observes the current queue baclogs and the current channel states, and selects a single lin for wireless transmission. This is a classic opportunistic scheduling scenario, where the networ scheduler can exploit nowledge of the current state of the time varying channels. It is well nown that maxweight scheduling policies are throughput optimal in such systems, in the sense that they provably stabilize all queues whenever the input rate vector is inside the networ capacity region. This stability result was first shown by Tassiulas and Ephremides in for the special case of O/OFF channels, and was later generalized to multi-rate transmission models and systems with power allocation However, the delay properties of max-weight scheduling are less understood. An average delay bound that is linear in is derived in 5 6. While this bound is tight in the case of correlated arrival and channel processes, it is widely believed to be loose for independent arrivals and channels. Michael J. eely is with the Electrical Engineering department at the University of Southern California, Los Angles, CA (web: mjneely). This material was presented in part at the Allerton Conference on Communication, Control, and Computing, Monticello, IL, Sept This material is supported in part by one or more of the following: the DARPA IT-MAET program grant W9F , the SF grant OCE 05034, the SF Career grant CCF In this paper, we focus on the special case of O/OFF channels, and show that the max-weight policy indeed yields average delay that is O() under independence assumptions. Thus, average delay does not grow with the networ size and hence is order optimal. While our previous queue grouping results in 7 also demonstrate that O() delay is possible, this is the first such result for the simpler class of maxweight policies. Specifically, we first show that for any input rate vector that is within a ρ-scaled version of the capacity region (where ρ represents the networ loading and satisfies 0 < ρ < ), the max-weight rule yields average delay that is less than or equal to c log(/( ρ)) ( ρ), where c is a constant that does not depend on ρ or. This is in comparison c ρ to the previous delay bound of developed for maxweight scheduling 5 6. ote that our new bound does not grow with, but has a worse asymptotic in ρ. We next present a different analysis that improves the delay bound to c log(/( ρ)) ρ for systems with f-balanced traffic rates (to be made precise in later sections). That is, if arrival rates are heterogeneous but are more balanced (so that the difference between the maximum arrival rate and the average arrival rate is sufficiently small), then order-optimal average delay is maintained while the delay asymptotic in ρ is improved. Finally, we consider systems with multi-rate capabilities. We first provide a delay bound that grows linearly with, similar to the bounds in 5 6 but with an improved coefficient. We then provide an example multi-rate system and show that its average congestion and delay must grow at least linearly with under any scheduling algorithm, due to many queues having a small number of residual pacets. This is an important example and demonstrates that the Θ() behavior of the multi-rate delay bound is fundamental and cannot be avoided, highlighting a significant difference between single-rate and multi-rate systems. It is nown that order-optimal delay requires queue-based scheduling. Indeed, it is shown in 7 that average delay in an -user downlin with time varying channels grows at least linearly with if queue-independent algorithms are used (such as round-robin or randomized schedulers). Related results are shown for pacet switches in 8, where a delay gap between queue-aware and queue-independent algorithms is developed. Delay optimal control laws for multi-user wireless systems are mostly limited to systems with special symmetric structure 9 0. Delay optimality results are developed in for a heavy traffic regime in the limit as the system loading ρ approaches. Recent results on exponents of the The value c is used here to easily express a delay scaling relationship, and represents a generic coefficient that does not depend on ρ or. The value c is not necessarily the same in all places it is used.

2 tail of delay distributions are provided in 3, and orderoptimal delay for greedy maximal scheduling with ρ a constant factor away from is considered in 4 5. The max-weight rule is also called the Longest Connected Queue (LCQ) scheduling rule in the special case of an O/OFF downlin. This policy was developed by Tassiulas and Ephremides in, where it was shown to support the full networ capacity region and to also be delay optimal in the special symmetric case when all arrival rates and O/OFF probabilities are the same for each lin. The fact that the actual average delay of LCQ in such symmetric cases is O() was recently proven in 0 (which shows that doubling the size of a symmetric system does not increase the average delay) and 7 (which uses a queue-grouped Lyapunov function to bound the average delay). Delay properties of variations of LCQ for symmetric Poisson systems are considered in 6 in the limit of asymptotically large. For asymmetric systems, it is shown in 7 that a different algorithm, called the Largest Connected Group (LCG) algorithm, yields O() average delay. However, the LCG algorithm requires some statistical nowledge to set up a queue-group structure. Hence, it is important to understand the delay properties of the simpler max-weight rule, which does not require statistical nowledge. In this paper, we combine the queue grouping concepts developed in 7 together with two novel Lyapunov functions to provide an order-optimal delay analysis of maxweight. The first Lyapunov function we use has a weighted sum of two different component functions, and is inspired by wor in 7 where a Lyapunov function with a similar structure is used in a different context. In the next section, we specify the networ model and review basic concepts concerning the networ capacity region. Section III proves our first delay result for the O/OFF channel model with general heterogeneous traffic rates inside the capacity region. Section IV provides our second bound (with a tighter asymptotic in ρ) for the case of heterogeneous traffic rates but under an f-balanced traffic assumption. Section V treats multi-rate systems. Section VI provides simulation results. II. SYSTEM MODEL Consider a multi-user wireless system with transmission lins. The system operates in slotted time with normalized slots t {0,,,...}. We assume that data is measured in units of fixed size pacets, and let A i (t) represent the number of pacets that arrive to lin i {,..., } during slot t. Each lin i maintains a separate queue to store this arriving data, and we let Q i (t) represent the number of pacets waiting for transmission over lin i. Let S i (t) represent the channel state for the ith channel during slot t. We assume that S i (t) is a non-negative integer that represents the current transmission rate (in units of pacets/slot) available over channel i if this channel is selected for transmission on slot t. For most of this paper, we consider the simple case of O/OFF channels, where S i (t) {0, } for all channels i {,...,} (multi-rate systems are treated in Section V). Define S(t) (S (t),..., S (t)) as the channel state vector. Let µ i (t) represent the control decision variable on slot t, given as follows: { Si (t), if channel i is selected on slot t µ i (t) 0, otherwise Define µ(t) (µ (t),..., µ (t)) as the vector of transmission decisions. We also call this the transmission rate vector, as it determines the instantaneous transmission rates over each lin (in units of pacets/slot), where the rate is either 0 or. The constraint that at most one channel is selected per slot translates into the constraint that µ(t) has at most one non-zero entry (and any non-zero entry i is equal to S i (t)). Define F(t) as the set of all such control vectors µ(t) that are possible for slot t, called the feasibility set for slot t. The queue dynamics for each queue i {,...,} are given as follows: Q i (t ) maxq i (t) µ i (t), 0 A i (t) () subject to the constraint µ(t) F(t) for all t. A. Traffic and Channel Assumptions We assume the arrival processes A i (t) are independent for all i {,...,}. Further, each process A i (t) is i.i.d. over slots with mean λ i E {A i (t)} and with a finite second moment E { A i (t) } <. Similarly, we assume channel processes S i (t) are independent of each other and i.i.d. over slots with probabilities PrS i (t) p i for i {,..., }. B. The etwor Capacity Region Suppose the networ control policy chooses a transmission rate vector every slot according to a well defined probability law, so that the queue states evolve according to (). Definition : A queue Q i (t) is strongly stable if: lim sup t t t τ0 E {Q i (τ)} < We say that the networ of queues is strongly stable if all individual queues are strongly stable. Throughout, we shall use the term stability to refer to strong stability. Define Λ as the networ capacity region, consisting of the closure of all arrival rate vectors λ (λ,..., λ ) for which there exists a stabilizing control algorithm. In it is shown that the capacity region Λ is the set of all rate vectors λ (λ,...,λ ) such that for each of the non-empty lin subsets L {,..., }, we have: λ i Π i L ( p i ) () i L This is an explicit description of the capacity region Λ. The following alternative implicit characterization is also useful for analysis (see 6 and references therein): Theorem : (Capacity Region Λ) The capacity region Λ is equal to the set of all (non-negative) rate vectors λ (λ,...,λ ) for which there exists a stationary randomized control policy that observes the current channel state vector

3 3 S(t) and chooses a feasible transmission rate vector µ(t) F(t) as a random function of S(t), such that: λ i E {µ i (t)} for all i {,...,} (3) where the expectation is taen with respect to the random channel vector S(t) and the potentially random control action that depends on S(t). It is easy to see that any non-negative rate vector that is entrywise less than or equal to a vector λ Λ is also contained in Λ. This follows immediately from Theorem by modifying the stationary randomized policy µ(t) that yields E {µ i (t)} λ i to a new policy ˆµ(t) by probabilistically setting each µ i (t) value to zero with an appropriate probability q i, yielding E {ˆµ i (t)} E {µ i (t)} ( q i ) E {µ i (t)}. It is also easy to show that the capacity region Λ is convex and compact (i.e., closed and bounded). Further, if E {S i (t)} > 0 for all i {,...,}, then Λ has full dimension of size and hence has a non-empty interior. C. The Max-Weight Scheduling Policy Given a rate vector λ interior to the capacity region Λ, a stationary, randomized, queue-independent policy could in principle be designed to stabilize the system, although this would require full nowledge of the traffic rates and channel state probabilities. However, it is well nown that the following queue-aware max-weight policy stabilizes the system whenever the rate vector is interior to Λ, without requiring nowledge of the traffic rates or channel statistics : Each slot t, observe current queue baclogs and channel states Q i (t) and S i (t) for each lin i, and choose to serve the lin i (t) {,..., } with the largest Q i (t)s i (t) product. This is also called the Longest Connected Queue policy (LCQ), as it serves the queue with the largest baclog among all that are currently O. The max-weight policy is very important because of its simplicity and its general stability properties. However, a tight delay analysis is quite challenging, and prior wor provides only a loose upper bound on average delay that is O(), i.e., linear in the networ size 5 6. It is shown in 7 that O() average delay is possible when both channels and pacet arrivals are independent across users and across timeslots, and when no traffic rate is larger than the average traffic rate by more than a specified amount. The O() delay analysis of 7 uses an algorithm called Largest Connected Group that is different from the max-weight policy and that requires more statistical nowledge to implement. In the following, we use the queue grouping analysis techniques of 7 to show that the simpler max-weight policy can also provide O() average delay, and does so for all traffic rates within a ρ-scaled version of the capacity region. However, the scaling in ρ is worse than that in 7. Section IV recovers the same ρ scaling as 7 under a similar f-balanced traffic assumption. III. DELAY AALYSIS FOR ARBITRARY RATES I Λ Consider the O/OFF channel model where each S i (t) is an independent i.i.d. Bernoulli process with PrS i (t) p i. Assume the arrival rate vector λ (λ,..., λ ) is interior to the capacity region Λ, so that there exists a value ρ such that 0 < ρ < and: λ ρλ (4) That is, λ is contained within a ρ-scaled version of the capacity region. The parameter ρ can be viewed as the networ loading, measuring the fraction the rate vector λ is away from the capacity region boundary. Define A tot (t) as the total pacet arrivals on slot t: A tot (t) A i (t) Define λ tot λ i as the sum pacet arrival rate. Because the sum of the entries of any rate vector in the capacity region Λ is no more than, we have by (4) that λ tot ρ. A. Important Parameters of Λ To analyze delay, it is useful to characterize the - dimensional capacity region Λ in terms of its size on subspaces of smaller dimension. To this end, define p min as the smallest channel O probability: p min min i {,...,} p i We assume that 0 < p min <. For each positive integer, define parameters µ sym and r as follows: µ sym ( p min) r ( p min ) Thus, r µ sym. The following lemma shall be useful. Lemma : For any positive integer and any probability p min > 0, we have µ sym > µsym. That is: ( p min) > ( p min) Proof: See Appendix D. Further, for {,..., }, let L represent a particular subset of lins within the lin set {,...,}. For each subset L, define L as an -dimensional vector that is in all entries i L, and zero in all other entries. Lemma : For each set L of size (for any integer such that ) we have: µ sym L Λ Furthermore, for each integer such that and for any set L that contains lins, we have µ sym L Λ. Proof: We first prove that µ sym L Λ. By (), it suffices to show that for any integer m such that m, the sum of any m non-zero components of µ sym L is less than or equal to r m. That is, it suffices to show that mµ sym r m. But this is equivalent to showing that µ sym µ sym m for m, which is true by Lemma. Finally, the fact that µ sym L Λ (for any integer such that ) follows because any rate vector with entries less than or equal to another rate vector in Λ is also in Λ. ote that r m Π i Lm ( p i ), where L m is any subset of m lins.

4 4 Thus, µ sym can be intuitively viewed as an edge size such that any -dimensional hypercube of this edge size (with dimensions defined along the orthogonal directions of any axes of R ) can fit inside the capacity region Λ. B. The O() Delay Bound for Arbitrary Traffic in Λ Suppose the LCQ algorithm is used together with a stationary probabilistic tie breaing rule in cases when multiple queues have the same weight. This allows the queueing system to be viewed as a stationary Marov chain. In this case, it is well nown that if the arrival rate vector is interior to the capacity region, then all queues are stable under LCQ, with a well defined steady state time average 6. The following O() delay bound for LCQ is given in 7: 3 W λ tot E{ A i (t) } λ tot λ i r ( ρ) where W represents the average delay in the system. The bound (5) also holds for arrival vectors A(t) that are i.i.d. over slots but with possibly correlated entries A i (t) on the same slot t. The next theorem demonstrates an improved O() bound in the case when all arrival processes A i (t) are independent. Theorem : (Delay Bound for LCQ) Consider the O/OFF channel model and assume processes A i (t) and S i (t) are independent and i.i.d. over slots. Assume that λ ρλ for some networ loading ρ such that 0 < ρ <. Let be any integer such that r > λ tot, that is: 4 (5) ( p min ) > λ tot (6) Then the max-weight (LCQ) policy for the O/OFF channel model stabilizes all queues and yields: Q i B θc ( ρ) where Q i is the time average number of pacets in queue i, and where the constants B θ, C, and θ are defined: λ tot B θ E { A i (t) } λ i θ { E Atot (t) } λ tot λ tot C θ { r r λ tot ( ρ) r (r λ tot ) ( ρ) if < /r if { ( ρ)(µ sym µsym ) r if < 0 if (7) (8) (9) (0) 3 The bound in 7 is of the form c/ǫ, where ǫ is any value such that λ ǫ Λ, where ǫ is a vector with all values equal to ǫ. The bound (5) follows by observing that ǫ ( ρ)r / satisfies λ ǫ Λ whenever λ ρλ. A similar c/ǫ bound is given in 5 6 for more general multi-rate systems. 4 ote that λ tot ρ <, and hence there is always a suitably large value such that (6) holds. By Little s Theorem, average delay W thus satisfies: B θ C W min λ tot ( ρ), B 0 λ tot r ( ρ) () where B 0 represents the value of B θ with θ 0, and the second expression in the above min, function is identical to the previous delay bound (5). The proof of Theorem is given in the next subsection. We note that the right term inside the min, operator in () is smaller in the case. The above bound can be minimized over all positive integers that satisfy r > λ tot. For a simpler interpretation of the bound that illuminates the fact that this is an O() delay result, note that because (ρ)/ > ρ λ tot, we can ensure that (6) holds by choosing to satisfy: ( p min ) ( ρ)/ Choosing as follows accomplishes this: max, log(/( ρ)) log(/( p min )) () Because λ tot ρ, it is not difficult to show that with this choice of, we have r λ tot ( ρ)/. Thus, in the case < we have: C r r (r λ tot) ( ρ) r r Because C /r for the case, we have that C O() (regardless of whether or not < ). Further, we have from () that is proportional to log(/( ρ)) but independent of. Finally, if arrival processes are independent so that E { A tot (t) } O(), we have B θ /λ tot O(). Therefore, the delay bound of () has the form: W min c log(/( ρ)) ( ρ), c ( ρ) (3) where c and c are constants that do not depend on ρ or. If itself is small, then the right expression in the above min, can be smaller than the left expression (i.e., the previous delay bound (5) can be the same as our new delay bound in the case when is small). However, if the loading ρ is held fixed as is scaled to infinity, then the left expression in the min, is always smaller and demonstrates O() average delay (see also simulations in Figs. and of Section VI). Thus, LCQ is order-optimal with respect to. However, the left delay bound has a worse asymptotic in ρ, and so it would be worse than the right bound in the opposite case when is fixed and ρ is scaled to. C. Lyapunov Drift Analysis To prove Theorem, it suffices to consider only the case <, as the delay bound in the opposite case is identical to the previous delay bound (5). Let Q(t) (Q (t),..., Q (t)) be the vector of queue baclogs. Define Q tot (t) as the sum queue baclog in all queues of the system: Q tot (t) Q i (t) (4)

5 5 Define the following Lyapunov function: L(Q(t)) ( Q i(t) θ j j(t)) Q (5) where θ is a positive constant to be determined later. Thus, L(Q(t)) Q i(t) θ Q tot(t). This Lyapunov function uses the standard sum of squares of queue length, and adds a new term that is the square of the total queue baclog. This new term incorporates the queue grouping concept similar to 7, and will be important in obtaining tight delay bounds. The technique of composing this Lyapunov function as a sum of two different quadratic terms weighted by a θ constant shall be useful in analyzing both stability and delay in two different modes of networ operation, and is inspired by a similar technique used in 7 to analyze stability in a very different context. Specifically, wor in 7 considers multihop networs with greedy maximal scheduling and achieves stability results when input rates are a constant factor (such as a factor of ) away from the capacity region boundary. Here, we consider a single-hop networ with time-varying channels, and obtain both stability and order-optimal delay results for all input rates inside the capacity region. The intuition on why this -part Lyapunov function allows a tight delay bound is as follows: The first term is a standard sum of squares of queue length, and ensures stability of the algorithm while creating a large negative drift when the number of nonempty queues is small. However, this term also has a relatively small negative drift when the number of non-empty queues is large, preventing O() delay analysis from this term alone. To compensate, the second term is a square of the sum of all queues, which does not significantly affect the drift of the first term when the number of non-empty queues is small, but creates a large negative drift to help the first term when the number of non-empty queues is large. The queue dynamics () can be rewritten as follows: Q i (t ) Q i (t) µ i (t) A i (t) (6) where µ i (t) minq i (t), µ i (t). Define µ tot (t) µ i(t), being either 0 or, and being if and only if the system serves a pacet on slot t. The dynamics for Q tot (t) are given by: Q tot (t ) Q tot (t) µ tot (t) A tot (t) (7) where A tot (t) A i(t). Let Q(t) be the stochastic queue evolution process for a given control policy. Define the one-step conditional Lyapunov drift as follows: 5 (Q(t)) E {L(Q(t )) L(Q(t)) Q(t)} (8) Lemma 3: The Lyapunov drift (Q(t)) for the O/OFF channel model satisfies: (Q(t)) E {B(t) Q(t)} Q i (t)e {µ i (t) λ i Q(t)} θq tot (t)e { µ tot (t) λ tot Q(t)} 5 Strictly speaing, correct notation should be (Q(t), t), as the drift could be from a non-stationary policy, although we use the simpler notation (Q(t)) as formal notation for the right hand side of (8). where µ i (t) and µ tot (t) correspond to the LCQ policy, and where B(t) is given by: B(t) µ tot(t) A i (t) A i (t) µ i (t) θ A tot(t) µ tot (t) µ tot (t)a tot (t) (9) Proof: (Lemma 3) See Appendix A. ow note that the LCQ algorithm chooses µ(t) F(t) on each slot t to maximize Q i(t)µ i (t), and hence: Q i (t)µ i (t) Q i (t)µ i (t) where µ (t) (µ (t),..., µ (t)) is any other feasible transmission rate vector in F(t). It follows that the above inequality is preserved when taing conditional expectations given the current Q(t) value. Plugging this result into the second term on the right hand side of the drift expression in Lemma 3 thus yields: (Q(t)) E {B(t) Q(t)} Q i (t)e {µ i (t) λ i Q(t)} θq tot (t)e { µ tot (t) λ tot Q(t)} (0) where µ (t) (µ (t),..., µ (t)) is any other feasible control action on slot t. ote that µ tot (t) in the above expression still corresponds to the LCQ policy. Let L(t) represent the number of non-empty queues on slot t, so that 0 L(t). Case (L(t) ): Suppose L(t), and let L(t) represent the set of non-empty queue indices. Recall that µ sym L(t) Λ (by Lemma ) and that λ/ρ Λ (by assumption that λ ρλ). By taing a convex combination of these two vectors and using convexity of the set Λ, it follows that: λ ( ρ)µ sym L(t) Λ () ow let µ (t) be the stationary randomized policy that maes decisions based only on the current channel state, and that yields: E {µ (t)} λ ( ρ)µ sym L(t) Such a policy exists by () and Theorem. Thus, for all i L(t) we have: E {µ i (t)} λ i ( ρ)µ sym () Using () in the drift inequality (0) and noting that Q i (t) 0 if i / L(t) yields: (Q(t)) E {B(t) Q(t)} θq tot (t)λ tot E {B(t) Q(t)} Q tot (t)( ρ)µ sym θλ tot Q i (t)( ρ)µ sym

6 6 Define ǫ as follows: It follows that: ǫ ( ρ)µ sym θλ tot (3) (Q(t)) E {B(t) Q(t)} ǫq tot (t) (4) Case (L(t) > ): Suppose L(t) >, and again let L(t) represent the set of non-empty queue indices. ote that λ/ρ Λ and µ sym Λ, where is the all vector. By convexity of Λ, the convex combination is also in Λ: λ ( ρ)µ sym Λ ow let µ (t) be the stationary randomized policy that maes decisions independent of queue baclog, and that yields for all i {,...,}: E {µ i (t)} λ i ( ρ)µ sym (5) Such a policy exists by Theorem. ote that when the number of non-empty queues is greater than, there is a pacet departure under the LCQ policy with probability at least one minus the product of the largest OFF probabilities: E { µ tot Q(t)} Π i ˆL ( p i ) r (6) where ˆL represents the set of lins with the smallest success probabilities. Plugging (5) and (6) into the drift inequality (0) yields: (Q(t)) E {B(t) Q(t)} Q tot (t)( ρ)µ sym θ(r λ tot ) To equalize the drift in both Case and Case, we choose θ to satisfy: Thus (using (3)): θ ǫ ǫ ( ρ)µ sym θ(r λ tot ) ( ρ)(µsym r ( ρ)µsym µsym ) λ tot µ sym (r λ tot ) r Recall that we have assumed < (as Theorem is trivially true if, as described at the beginning of this subsection). Thus, we have µ sym > µsym (by Lemma ), and so we indeed have θ > 0. Further, because r > λ tot, we have that ǫ > 0. Therefore, the drift inequality (4) holds in both Case and Case (and hence holds for all t and all Q(t)). We now use the following well nown Lyapunov drift lemma (see, for example, 6 for a proof): Lemma 4: (Lyapunov Drift 6) If the drift (Q(t)) of a non-negative Lyapunov function satisfies the following for all t and all Q(t): (Q(t)) E {B(t) Q(t)} ǫe {f(t) Q(t)} for some stochastic processes B(t), f(t), and some constant ǫ > 0, then: f B/ǫ where f B t lim sup t t limsup t t τ0 t E {f(τ)} E {B(τ)} τ0 Using this Lyapunov drift lemma in (4) (using f(t) Q tot (t)) yields: Q tot B/ǫ We note that because the system evolves according to a Marov chain with a countably infinite state space, the time averages are well defined (so that the limsup can be replaced by a regular limit). Further, using the fact that t lim t t τ0 E { µ i(τ)} λ i, the value of B can be seen to equal the value B θ defined in (7), proving Theorem. IV. A TIGHTER BOUD FOR f -BALACED TRAFFIC Here we present a tighter bound on average baclog and delay of the LCQ algorithm for the O/OFF channel model. Our bound in this section is of the form c log( ρ )/( ρ), which is still O() with respect to the networ size, but yields a better asymptotic in ρ. Unfortunately, our analysis does not hold for all rate vectors λ inside the capacity region Λ. Rather, we mae the following assumption about a more balanced traffic rate vector. Let λ (λ,..., λ ), and without loss of generality assume that λ i > 0 for all i {,..., } (else, we can redefine to be the number of lins with non-zero rates). Define λ tot λ i and λ av λ tot /. We say that λ has f-balanced rates if there is a constant f such that: λ i λ av f for all i {,...,} (7) That is, λ is f-balanced if no individual traffic rate is more than an amount f above the average rate λ av. Clearly any uniform traffic rate vector is f-balanced for f 0. However, this definition of f-balanced rates also captures a large class of heterogeneous arrival rate vectors. We shall prove our delay results under the assumption that f is suitably small. A similar assumption is used in 7, and our delay analysis relies heavily on the queue-grouping techniques used there. A. The Queue-Grouped Lyapunov Function Fix an integer such that. Define ˆ as the smallest multiple of that is larger than or equal to : ˆ / (8) ow define a new rate vector ˆλ (λ,..., λ, 0, 0,..., 0), where the last ˆ entries are zero. Define ˆ fictitious queues for these last dimensions (these queues always have zero baclog, but shall be convenient to define for counting purposes). Define G as the set of all possible partitions of the lin set {,..., ˆ} into disjoint sets, each ˆ/ lins. Let g G denote a as the collection is equal with an equal size of particular partition, and define L (g),..., L(g) of sets corresponding to g (so that the union L(g)

7 7 to {,..., ˆ}, and the intersection L (g) n all m n, where m, n {,...,}). L (g) m is empty for For a particular partition g, define (t) as the sum of all queue baclogs in the th set of g: (t) Q i (t) i L (g) Define the following queue-grouped Lyapunov function: L(Q(t)) g G ( (t)) (9) This is similar to the Lyapunov function of 7, with the exception that it sums over all possible partitions into disjoint groups. For intuition, we note that the f-balanced traffic assumption allows the Largest Connected Group (LCG) argument of 7 to proceed on any set of disjoint groups. However, once we fix a particular group, minimizing the drift gives rise to the LCG algorithm rather than the max-weight LCQ algorithm. Changing the Lyapunov function by summing over all possible disjoint groups yields a similar negative drift as in LCG, but the symmetry induced by summing over all groups remarably maes the drift minimizing algorithm the LCQ algorithm (rather than LCG). Define A (g) (t) and µ(g) (t) as the sum arrivals and departures from the th group of the partition g: A (g) (t) A i (t) µ (g) i L (g) (t) i L (g) µ i (t) The dynamics for the th group of partition g thus satisfy: (t ) Q(g) (t) µ(g) (t) A(g) (t) (30) Define the Lyapunov drift (Q(t)) as before (given in (8)). Lemma 5: For a general scheduling policy, the Lyapunov drift satisfies: (Q(t)) E {C(t) Q(t)} g G (t)e { } µ (g) (t) λ(g) Q(t) where λ (g) λ i L (g) i, and where C(t) is defined: C(t) µ (g) (t) A(g) (t) µ (g) (t)a(g) (t) g G (3) Proof: The proof is similar to the proof of Lemma 3. Specifically, note from (30) that: (t ) (t) A(g) (t) (t) µ (g) µ (g) (t)a(g) (t) Q(g) (t) µ(g) (t) A(g) (t) where we have used the fact that µ (g) (t) µ (g) (t). The result follows by summing over all and all groups, and taing conditional expectations. Remarably, we next show that the max-weight LCQ algorithm for this O/OFF channel model minimizes the final term in the right hand side of the above drift expression. Lemma 6: (Max Weight Matching) Every slot t, the LCQ algorithm chooses a transmission rate vector µ(t) F(t) that maximizes the following expression over all alternative feasible transmission rate vectors: g G (t) µ(g) (t) Proof: See Appendix B. It follows that we can replace the variables µ (g) (t) in the final term of the drift expression in Lemma 5, which correspond to the LCQ policy, with variables µ (g) (t) that correspond to any other feasible rate vector µ (t) F(t), while creating an inequality relationship: (Q(t)) E {C(t) Q(t)} { (t)e µ (g) (t) λ (g) }(3) Q(t) g G The drift inequality (3) is quite subtle: It is defined in terms of any other single feasible rate vector µ (t) (where this vector does not depend on the partition g). ote that the variables (t) are defined for different partitions g G, but for each particular g these variables are still derived from the same vector µ (t). They are derived from µ (t) by summing the components of this rate vector that have non-empty queues over the dimensions that correspond to the groups within the particular partition g. µ (g) B. Optimizing the Drift Bound Here we manipulate the sum in the right-hand side of (3) to yield a useful drift bound. Lemma 7: For any vector λ (λ,...,λ ˆ), if there is a value β such that 0 < β < such that for all i {,...,} we have: then: g G λ i λ tot ˆ β( ρ) (33) (t)λ(g) Q tot (t) G λ tot zβ( ρ) where G is the cardinality of G, Q tot (t) is the total sum baclog (defined in (4)), and z is defined: z ( /)/( / ˆ) (34) Proof: The proof of Lemma 7 follows from simple counting arguments, and is given in Appendix C. ote that λ tot / ˆ λ tot / with approximate equality when is large (so that ˆ/ ). The constraints (33) imply that λ is f-balanced with f β( ρ)/. Lemma 8: There exists a single randomized strategy that observes queue baclogs and channel states for slot t and chooses µ (t) F(t) such that: g G (t)e { µ (g) (t) Q(t) } Q tot (t) G r

8 8 where r ( p min ). Proof: See Appendix C. Using Lemmas 7 and 8 in the drift inequality (3) yields: Lemma 9: If λ ρλ (for 0 < ρ < ) and if (33) is satisfied for all i {,...,}, then: (Q(t)) E {C(t) Q(t)} Q tot (t) G r λ tot zβ( ρ) (35) Lemma 9 leads immediately to the delay theorem stated in the next subsection. C. An Improved Delay Bound for f-balanced Traffic Theorem 3: (Delay Bound for O/OFF Channels with f- Balanced Traffic) Suppose λ ρλ for 0 < ρ <. Let be the smallest integer that satisfies r ( ρ)/, that is: ( p min ) ( ρ)/ (36) Suppose that, and the f-balanced traffic constraints (33) are satisfied for some value β such that 0 β < /(z), where z ( /)/( / ˆ) (note that z ). If the maxweight (LCQ) policy is used on this O/OFF channel model, then average queue occupancy satisfies: Q tot D ( ρ)( zβ), W (D/λ tot) ( ρ)( zβ) (37) where D is defined: D λtot E { A tot (t) } Further, in the special case when is a multiple of, and when traffic is uniform and Poisson with λ i λ tot / for all i, we have β 0 and: 6 Q tot λ tot λ tot ρ, W λ tot ρ ote that the constraint (36) is satisfied by: log( ρ ) log(/( p min )) Therefore, is independent of, and is proportional to log( ρ ). Assuming that traffic streams are independent, so that E { A tot (t) } O(), implies that D O(). Thus the delay bound gives W c log(/( ρ)) ρ (where c is a constant independent of ρ and ), being independent of the networ size and having an asymptotic in ρ that is better than that of Theorem. Proof: (Theorem 3) Because λ ρλ, we have λ tot ρ (as the maximum sum rate is at most r ). The assumption on r in (36) thus implies: r λ tot zβ( ρ) ( ρ)( zβ) 6 These bounds for symmetric Poisson traffic are obtained from the last line of the proof of Theorem 3, which gives a slightly smaller bound than that achieved by plugging β 0, E A tot(t) λ tot λ tot into (37). The above value is strictly positive because zβ < /. Using the drift inequality (35) directly in the Lyapunov Drift Lemma (Lemma 4) yields: Q tot C G ( ρ)( zβ) Using the definition of C(t) in (3) and the fact that the system is stable (so the long term departure rate is equal to λ tot ) yields: C G λ tot { E A (g) (t)} (λ (g) ) g G λ tot G E { Atot (t) } G D The above bound on C proves the first part of the theorem. The second part, for uniform Poisson traffic, follows { by the } above equality for C (without the bound), using E A (g) (t) λtot { } and E A (g) (t) λ tot λtot for all g,. V. MULTI-RATE TRASMISSIO MODELS ow suppose that for each channel i {,..., }, the states S i (t) are non-negative integers bounded by a finite integer µ i,max, where µ i,max represents the maximum transmission rate over channel i. 7 That is, we have: S i (t) {0,,..., µ i,max } for all t and all i {,...,} We assume that µ i,max > 0 for all i. The queueing dynamics are governed by (). The capacity region Λ is nown to be equal to the set of all rate vectors that can be achieved via a stationary, randomized, queue-independent algorithm that chooses µ (t) as a potentially random function of only the current S(t) vector 6. The max-weight algorithm in this case is the algorithm that observes queue baclogs and channel states every slot and selects the lin i {,...,} with the largest value of Q i (t)s i (t) (breaing ties arbitrarily). Suppose the arrival rate vector satisfies λ ρλ for some loading value ρ such that 0 < ρ <. The analysis in 5 6 uses a standard Lyapunov function, given by the sum of the squares of queue baclog, to show the max-weight algorithm for a general downlin has average delay upper bounded by c/( ρ), where c is a constant that is independent of and ρ. We first present a modified version of that prior bound, which has the same structure but uses our particular µ i,max notation and improves the c coefficient: Lemma 0: Suppose A(t) is i.i.d. over slots with E {A(t)} λ, and that the channel state vector S(t) (S (t),..., S (t)) is also i.i.d. over slots. Suppose that λ ρλ for some value ρ that satisfies 0 < ρ <. Then the system 7 For consistency, we continue to wor in integer units of pacets. The analysis does not significantly change if S i (t) values are viewed as nonnegative real numbers with units of bits/slot.

9 9 is stable under the max-weight algorithm and has an average delay bound given by: λ tot E{ } A i 3 λ tot λ i W ( ρ)µ sym min λ iµ i,max λ tot, Ŝ λ tot ( ρ)µ sym (38) where Ŝ is defined: { } Ŝ E max S i(t) i {,...,} and where µ sym is defined as the largest value such that (µ sym /, µ sym /,...,µ sym /) Λ. Further, in the case when all processes S i (t) are independent and satisfy PrS i (t) µ i,max > 0, we can bound µ sym as follows: µ sym ˆµ( ( p min ) ) where ˆµ and p min are defined: ˆµ min i {,...,} µ i,max p min min i {,...,} PrS i(t) ˆµ Proof: The proof uses the Lyapunov function L(Q(t)) Q i(t), as in 5 6, but provides a simple modification of the argument to yield a tighter bound (given in Appendix E for completeness). We note that the above delay bound holds also in the case when µ i,max for some values i, but when the second moment of transmission rates is finite (so that Ŝ is finite). The above delay bound has the structure c/( ρ), and holds even if arrival and channel vectors A(t) and S(t) have entries that are correlated over the different lins i {,..., }. A similar argument can be used to show stability with the same structural delay bound ĉ/( ρ) for the modified maxweight policy that chooses the lin i {,...,} with the largest Q i (t)minq i (t), S i (t) value. This modified policy can sometimes provide smaller empirical average delay than the original max-weight policy, although its resulting analytical delay bound has a slightly worse coefficient ĉ c (this modified policy is equivalent to the original max-weight policy in the case of O/OFF channels with µ i,max for all i). Similar to the O/OFF case, one might suspect that for this multi-rate system, average delay that is independent of can be achieved when arrival and channel processes are independent over each channel. However, the next subsection presents an important example that shows this is not the case. 8 8 We note that our original pre-print of this paper in 8 incorrectly claimed that multi-rate systems also have delay that is independent of. The mistae in 8 arose when plugging the equation from Lemma 0 of that paper into equation (33) of that paper. Plugging one equation into the other implicitly assumed that the sum queue baclog in queues with at least µ max pacets is the same as the total queue baclog in the system. This is true when µ max, but is not true in general as it neglects the residual pacets in queues with fewer than µ max pacets. A. An example showing necessity of O() delay Here we present an example showing that the average number of queues that have at least one pacet but fewer than µ i,max pacets must be linear in, which necessarily maes the average delay of any scheduling policy grow at least linearly with. Consider a system with queues with symmetric channels and traffic. Assume that 3 and suppose that all arrival processes A i (t) are independent and Bernoulli with PrA i (t) 3/ for all i {,..., } (so that λ i 3/ for all i, and λ tot 3 pacets/slot). ow suppose that all channels have µ i,max 5, and channel state processes are i.i.d. with PrS i (t) 5 /, PrS i (t) 0 / for all i {,..., }. The largest symmetric rate in the capacity region of this system is µ sym / 5( (/) )/, and hence the arrival rate vector is inside the capacity region and has ρ given by: ρ 3 5( (/) ) ote that ρ < for 3, and ρ is approximately 3/5 for large. Here we show that under any scheduling policy, in steady state the average number of non-empty queues in this system must be linear in. Specifically, consider any scheduling policy, and let Z(t) represent the number of nonempty queues on slot t. For simplicity, we assume that Z(t) has a well defined steady state under the scheduling policy. The intuition behind our proof is that Z(t) is formed from Z(t) by adding the number of new non-empty queues created and subtracting any non-empty queue that becomes empty. The number of non-empty queues subracted can be at most (as we can serve at most one channel per slot), while the average number of new non-empty queues added is more than one whenever Z(t) < /. Lemma : Consider any scheduling policy for which Z(t) has a well defined steady state distribution. Then for the system above (with λ i 3/ and µ i,max 5 for all i {,..., }) we have that in steady state: PrZ(t) / /3 and hence E {Z(t)} /6. That is, the average number of non-empty queues is at least /6, and hence the average number of pacets in the system is at least /6. Proof: Define (t) Z(t ) Z(t) as the change in Z(t) from one slot to the next. Let t be a time at which the system is in steady state. We thus have E { (t)} 0. On the other hand, we have the following: E { (t) Z(t) /} (39) E { (t) Z(t) < /} λ i / / (40) where (39) follows because the drift cannot be less than on any slot (as at most one non-empty queue can become an empty queue), and (40) holds because, given that Z(t) < /, the average number of new non-empty queues that are created on slot t is equal to the average number of new arrivals to the

10 0 empty queues, which is at least λ i (/). It follows that: 0 E { (t)} E { (t) Z(t) /}PrZ(t) / E { (t) Z(t) < /}( PrZ(t) /) ( )P rz(t) / (/)( PrZ(t) /) Therefore PrZ(t) / /3, completing the proof. Average Sum Queue Baclog Average Baclog versus for Symmetric Traffic and rho 0.8 Previous O() Bound First O() Bound Second O() Bound Simulation VI. SIMULATIOS Here we present simulations for the O/OFF system with independent channel and arrival processes. We assume that PrS i (t) O / for all i {,...,}. We first consider symmetric Bernoulli arrivals, so that λ i λ for all i, where λ is chosen so that λ ρλ with ρ 0.8. We simulated the system over 0 6 slots for values of between 3 and 300. The resulting simulated queue averages are shown in Fig., together with the two O() bounds and the previous O() bound. ote that the previous O() bound is a considerable overestimate of queue baclog. Our new bounds do not grow with, and our second O() bound (derived for f-balanced traffic rates) is indeed tighter than the first O() bound (for 9), although it applies only to f-balanced traffic while the first bound applies to any traffic rates in ρλ. However, there is still a significant gap (roughly a factor of 0 in this example) between our tightest bound and the simulated value. We next consider heterogeneous traffic rates implemented on the same O/OFF system. We assume that is odd, and choose rates λ i given as follows: λ for i {,..., ( )/} λ i λ for i {( )/,..., } 4λ for i where λ is chosen so that λ ρλ for ρ 0.8. The results are shown in Fig.. ote that we plot only the first O() bound (for heterogeneous traffic) in this case, although the f- balanced traffic assumption also applies in this case when is sufficiently large. Simulations of the multi-rate system example in Section V-A were also conducted, and it was verified that average baclog indeed grows linearly with due to the residual pacets in queues i that have fewer than µ i,max pacets (simulation plots omitted for brevity). However, it was observed in the simulations that the total baclog due to queues with at least µ i,max pacets is O(). This suggests that, although the total average baclog in multi-rate systems may have a fundamental O() term due to residual pacets, the average baclog may be O() after a term of at most (µ i,max ) is subtracted out. VII. COCLUSIOS We have presented an improved delay analysis for the maxweight scheduling algorithm. For O/OFF channels, maxweight is equivalent to Longest Connected Queue (LCQ), and yields average delay that is order-optimal, being independent Fig.. Simulation and Bounds for the O/OFF system with symmetric traffic and ρ 0.8. Average Sum Queue Baclog Average Baclog versus for Heterogeneous Traffic and rho 0.8 Previous O() Bound O() Bound Simulation Fig.. Simulation and Bounds for the O/OFF system with heterogeneous traffic and ρ 0.8. of the networ size. If an f-balanced traffic assumption holds, average delay was shown to maintain independence of while allowing an improved asymptotic in ρ. For multirate channels, a delay bound of O() applies. Conversely, it is shown for a simple multi-rate example that, unlie O/OFF channels, average baclog must be at least linear in due to residual pacets. Our delay analysis maes use of the technique of queue grouping. The particular Lyapunov functions introduced for this delay analysis are powerful and may be useful in other contexts. APPEDIX A PROOF OF LEMMA 3 Here we prove Lemma 3. Define (Q(t)) and (Q(t)) as the conditional drift for the sum of squares term and the square of queue baclog term, respectively, so that (Q(t)) (Q(t))θ (Q(t)). Squaring (6) and using the fact that µ i (t) µ i (t) and Q i (t) µ i (t) Q i (t)µ i (t) (because µ i (t)

11 {0, }, and µ i (t) µ i (t) if Q i (t) > 0): yields: Q i(t ) Q i(t) (A i(t) µ i (t)) Q i (t)(µ i (t) A i (t)) Qi (t) A i (t) µ i (t) A i (t) µ i (t) Q i (t)(µ i (t) A i (t)) Therefore: where (Q(t)) E {B (t) Q(t)} Q i (t)e {µ i (t) λ i Q(t)} B (t) Ai (t) µ i (t) A i (t) µ i (t) Similarly: Q tot(t ) Qtot (t) µ tot (t) A tot (t) µ tot (t)a tot (t) Therefore: (Q(t)) Q tot (t)( µ tot (t) A tot (t)) E {B (t) Q(t)} Q tot (t)e { µ tot (t) λ tot Q(t)} where µ tot (t) µ tot (t) (because it is either 0 or ), and B (t) µtot (t) A tot (t) µ tot (t)a tot (t) Summing (Q(t)) and θ (Q(t)) and noting from (9) that B(t) B (t) θb (t) yields the result of Lemma 3. APPEDIX B PROOF OF LEMMA 6 Define the integer m ˆ/. Here we prove Lemma 6. Given a particular queue baclog vector Q(t), the LCQ algorithm maximizes the expression ˆ Q i(t)µ i (t) over all µ(t) F(t). We now show that this also maximizes the expression given in Lemma 6. To this end, we have: g G (t) µ(g) (t) g G ˆ ˆ (t) µ i (t)q i (t) G µ i (t) j i i L (g) µ i (t) Q j (t) G (m ) ˆ where the final equality holds because lin i is in every group that multiplies the µ i (t) term, and all other lins multiply this term the same number of times (by group symmetry). The above also uses the fact that (by symmetry) the number of group partitions for which a particular lin j is in the same group as lin i is equal to the total number of partitions multiplied by the probability that a randomly chosen partition includes i and j in the same group. Define the above expression as f(t) for simplicity. Therefore: f(t) ˆ ˆ µ i (t)q i (t) G ( m ˆ ) ˆ µ i (t) Q j (t) G m ˆ (4) j The µ i (t) values in the expression for f(t) are the only ones affected by the control action on slot t. The final term on the right hand side is given by i µ i(t) (the total departures on slot t) multiplied by a non-negative constant. This final term is maximized by any wor conserving policy that always transmits a pacet when there is a non-empty connected queue. The first term on the right hand side is a non-negative constant multiplied by the term i µ i(t)q i (t). But note that µ i (t)q i (t) µ i (t)q i (t), and thus the LCQ policy maximizes this first term. As LCQ is wor conserving, it also maximizes the second term, and thus maximizes f(t), proving Lemma 6. APPEDIX C PROOF OF LEMMAS 7 AD 8 Proof: (Lemma 7) Define the integer m ˆ/. Using a counting argument similar to that of Appendix B (compare with (4)), we have that for any vector λ (λ,...,λ ˆ): g G (t)λ(g) ˆ Using the bound on λ i given in (33) yields: g G (t)λ(g) Q tot (t)λ tot G Q i (t)λ i G ( m ˆ ) Q tot (t)λ tot G m ˆ (4) ˆ m ˆ( ˆ ) m ˆ Q tot (t) G ( m )β( ρ)/ ˆ The result of Lemma 7 follows by using the identities: ˆ m ˆ( ˆ ) m ˆ m z ˆ (43) Proof: (Lemma 8) Let L(t) represent the number of nonempty queues on slot t. If L(t) 0, then Q tot (t) 0 and the result is trivial. ow suppose that L(t) l, where l {,,..., }. Define (l,...,l ˆ) to be a 0/ vector with l i if and only if Q i (t) > 0. Define l (g) to be the number of non-empty queues in the th group of partition g. Consider the following randomized policy for µ (t) F(t): First observe

12 all channel states S i (t) for non-empty queues i, and define new channel states Ŝi(t) as follows: If S i (t) 0 (OFF), assign Ŝ i (t) 0. If S i (t) (O), independently assign Ŝi(t) with probability p min /p i (this is a valid probability because p min p i ). It follows that the new channel state vector Ŝ(t) has independent and symmetric O probabilities p min. ow independently, randomly, and uniformly choose a queue to serve over all non-empty queues i with Ŝi(t). It follows that for all non-empty queues i we have: E { µ i (t) Q(t)} ( p min) l i L (g) l r l l Further, for any g G and any {,...,} we have: { } E µ (g) (t) Q(t) E { µ i (t) Q(t)} l (g) r l l Using this equality gives: g G { } (t)e µ (g) (t) Q(t) r l l g G (t)l(g) (44) ow note that the l (g) values are structurally similar to the λ (g) values, and hence (similar to (4)) we have (using Q i (t)l i Q i (t) and l tot l): g G (t)l(g) Using this in (44) yields: g G r l l Q tot(t) G r l Q tot (t) G ˆ Q i (t) G ( m ˆ ) Q tot (t)l G m ˆ { } (t)e µ (g) (t) Q(t) ˆ m ˆ l(m ) ˆ (m ) ˆ( ˆ ) m ˆ (45) r l Q tot (t) G / (46) where the last equality holds by (43). The above holds for L(t) l {,..., }. Suppose now that l. In this case we have r l r, proving the result of Lemma 8 for l. Consider now the final case where l {,..., }. Then from (45) we have: g G { } (t)e µ (g) (t) Q(t) Using the fact that r l l r r r l l Q tot(t) G (47) yields the result. APPEDIX D PROOF OF LEMMA Here we prove that µ sym > µsym. Specifically, we show that if p is a value such that 0 < p, then for any positive integer we have: ( p) > ( p) (48) To show this, note that it is trivially true for the case p. In the opposite case where 0 < p <, we can multiply (48) by ( ) and rearrange terms to see that the inequality is equivalent to the following: ( p) p( p) < (49) Thus, it suffices to prove that (49) is true. To this end, we have: ( p) p( p) < ( p) p( p) ( ) p i ( p) i i i0 (( p) p) where the first (strict) inequality holds because 0 < p < and hence ( p) < ( p). This establishes (49) and completes the proof of Lemma. APPEDIX E PROOF OF LEMMA 0 The queueing dynamics are given by Q i (t ) Q i (t) µ i (t) A i (t), where µ i (t) minµ i (t), Q i (t). Using the Lyapunov function L(Q(t)) Q i(t) and performing a standard quadratic drift computation (see, for example, 6), it is not difficult to show the drift satisfies: (Q(t)) E { A i (t) } E {λ i µ i (t) µ i(t) } Q(t) λ i Q i (t) Q i (t)e {µ i (t) Q(t)} Q i (t)e {µ i (t) µ i (t) Q(t)} By definition of µ i (t), we have: Q i (t)(µ i (t) µ i (t)) µ i (t)µ i (t) µ i (t) Hence: (Q(t)) E { A i (t) } E {λ i µ i (t) µ i(t) minµ i,max µ i (t), µ i (t) µ i (t) } Q(t) λ i Q i (t) Q i (t)e {µ i (t) Q(t)} E { min µ i,max µ i (t), µ i (t) Q(t) } (50)